Higher-Order Discretization of Diffusion Terms in Residual-Distribution Methods

We discuss various higher-order discretization methods for diffusion terms in residual-distribution methods. Categorizing methods into two types: node-based (Galerkin methods) and cell-based (residual-distribution methods), we begin with the description of the basic low-order methods. We then consider two different approaches to extend these methods to higher-order: reconstruction and higher-order elements. Applying these to the low-order methods, we derive and discuss various higher-order methods for diffusion, with more emphasis on the cell-based methods, especially the methods based on the first-order system by which we can avoid discretizing second-derivatives. Numerical results are given for a simple test problem to demonstrate the accuracy of the derived schemes. In particular, it is shown that employing the first-order system we can achieve fourth-order accuracy with P2 elements. Finally, we discuss an issue for integrating these diffusion schemes with advection schemes, comparing two possible ways to construct higher-order advection-diffusion schemes.